Article Contents

2023, Volume 17, Issue 1: 78-97. Doi: 10.3934/amc.2021072

This issue Previous Article Minimal codewords arising from the incidence of points and hyperplanes in projective spaces Next Article Upper bounds on the length function for covering codes with covering radius

$ R $

and codimension

$ tR+1 $

An algorithmic approach to entanglement-assisted quantum error-correcting codes from the Hermitian curve

1.
Department of Mathematical Sciences, Aalborg University, Skjernvej 4A, 9220 Aalborg Øst, Denmark
2.
IMUVA-Mathematics Research Institute, Universidad de Valladolid, Paseo Belén 7, 47011 Valladolid, Spain
3.
Department of Electrical Engineering, Federal University of Campina Grande, Rua Aprígio Veloso 882, 58190-970, Campina Grande, Paraíba, Brazil
4.
School of Science and Technology, University of Camerino, I-62032 Camerino, Italy
5.
INFN, Sezione di Perugia, Via A. Pascoli, 06123 Perugia, Italy

^*Corresponding author: René B. Christensen
^*Corresponding author: René B. Christensen

Received: May 2021

Revised: November 2021

Early access: January 2022

Published: December 2022

This work was supported in part by Grant PGC2018-096446-B-C21 funded by MCIN/AEI/10.13039/501100011033 and by \ERDF A way of making Europe", by Grant RYC- 2016-20208 funded by MCIN/AEI/10.13039/501100011033 and by \ESF Investing in your future", and by the European Union's Horizon 2020 research and innovation programme, under grant agreement QUARTET No 862644

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Abstract

We study entanglement-assisted quantum error-correcting codes (EAQECCs) arising from classical one-point algebraic geometry codes from the Hermitian curve with respect to the Hermitian inner product. Their only unknown parameter is $ c $, the number of required maximally entangled quantum states since the Hermitian dual of an AG code is unknown. In this article, we present an efficient algorithmic approach for computing $ c $ for this family of EAQECCs. As a result, this algorithm allows us to provide EAQECCs with excellent parameters over any field size.

Keywords:

Quantum error-correcting code,
CSS construction,
entanglement-assisted quantum error-correcting code,
Hermitian code.

Mathematics Subject Classification: 94B65, 81P70, 94B05.

Citation:

Full Text(HTML)

Table 1. Normalized reductions for $ q = 3 $ and $ m = 22 $

$ i $	$ f_i $	$ \mathfrak{r}'(f_i^q) $	$ \mathfrak{r}(f_i^q)=\mathfrak{n}(\mathfrak{r}(f_i^q)) $
$ 1 $	$ 1 $	$ 1 $	$ 1 $
$ 2 $	$ x $	$ x^3 $	$ x^3 $
$ 3 $	$ y $	$ x^4 $ $ + $ $ 2y $	$ x^4 $ $ + $ $ 2y $
$ 4 $	$ x^2 $	$ x^6 $	$ x^6 $
$ 5 $	$ xy $	$ x^7 $ $ + $ $ 2x^3y $	$ x^7 $ $ + $ $ 2x^3y $
$ 6 $	$ y^2 $	$ x^8 $ $ + $ $ x^4y $ $ + $ $ y^2 $	$ x^8 $ $ + $ $ x^4y $ $ + $ $ y^2 $
$ 7 $	$ x^3 $	$ x $	$ x $
$ 8 $	$ x^2y $	$ 2x^6y $ $ + $ $ x^2 $	$ x^6y $ $ + $ $ 2x^2 $
$ 9 $	$ xy^2 $	$ x^7y $ $ + $ $ x^3y^2 $ $ + $ $ x^3 $	$ x^7y $ $ + $ $ x^3y^2 $ $ + $ $ x^3 $
$ 10 $	$ x^4 $	$ x^4 $	$ x^4 $
$ 11 $	$ x^3y $	$ x^5 $ $ + $ $ 2xy $	$ x^5 $ $ + $ $ 2xy $
$ 12 $	$ x^2y^2 $	$ x^6y^2 $ $ + $ $ x^6 $ $ + $ $ x^2y $	$ x^6y^2 $ $ + $ $ x^6 $ $ + $ $ x^2y $
$ 13 $	$ x^5 $	$ x^7 $	$ x^7 $
$ 14 $	$ x^4y $	$ x^8 $ $ + $ $ 2x^4y $	$ x^8 $ $ + $ $ 2x^4y $
$ 15 $	$ x^3y^2 $	$ x^5y $ $ + $ $ xy^2 $ $ + $ $ x $	$ x^5y $ $ + $ $ xy^2 $ $ + $ $ x $
$ 16 $	$ x^6 $	$ x^2 $	$ x^2 $
$ 17 $	$ x^5y $	$ 2x^7y $ $ + $ $ x^3 $	$ x^7y $ $ + $ $ 2x^3 $
$ 18 $	$ x^4y^2 $	$ x^8y $ $ + $ $ x^4y^2 $ $ + $ $ x^4 $	$ x^8y $ $ + $ $ x^4y^2 $ $ + $ $ x^4 $
$ 19 $	$ x^7 $	$ x^5 $	$ x^5 $
$ 20 $	$ x^6y $	$ x^6 $ $ + $ $ 2x^2y $	$ x^6 $ $ + $ $ 2x^2y $

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Table 2. Algorithm 1 for $ q = 3 $

$ f_i $	$ \nu(f_i) $	$ \mathfrak{r}(f_i^q) $	$ \nu(\mathfrak{r}(f_i^q)) $	$ \phi_i $	$ \nu(\phi_i) $
$ 1 $	0	$ 1 $	0	$ 1 $	0
$ x $	3	$ x^3 $	9	$ x^3 $	9
$ y $	4	$ x^4+2y $	12	$ x^4+2y $	12
$ x^2 $	6	$ x^6 $	18	$ x^6 $	18
$ xy $	7	$ x^7+2x^3y $	21	$ x^7+2x^3y $	21
$ y^2 $	8	$ x^8+x^4y+y^2 $	24	$ x^8+x^4y+y^2 $	24
$ x^3 $	9	$ x $	3	$ x $	3
$ x^2y $	10	$ x^6y+2x^2 $	22	$ x^6y+2x^2 $	22
$ xy^2 $	11	$ x^7y+x^3y^2+x^3 $	25	$ x^7y+x^3y^2+x^3 $	25
$ x^4 $	12	$ x^4 $	12	$ y $	4
$ x^3y $	13	$ x^5+2xy $	15	$ x^5+2xy $	15
$ x^2y^2 $	14	$ x^6y^2+x^6+x^2y $	26	$ x^6y^2+x^6+x^2y $	26
$ x^5 $	15	$ x^7 $	21	$ x^3y $	13

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Table 3. Examples of code's parameters and comparative analysis by means of coding bounds

Parameters	Singleton defect	Exceeding GV
$ [[27, 1, 19; 16]]_3 $	6	✔
$ [[27, 4, 16; 13]]_3 $	6	✔
$ [[27, 13, 7; 4]]_3 $	6	✔
$ [[27, 16, 4; 1]]_3 $	6	✔
$ [[64, 5, 42; 35]]_4 $	12	✔
$ [[64, 16, 30; 22]]_4 $	12	✔
$ [[64, 35, 12; 3]]_4 $	10	✔
$ [[64, 39, 8; 1]]_4 $	12	✔
$ [[125, 1,101; 96]]_5 $	20	✔
$ [[125, 9, 91; 84]]_5 $	20	✔
$ [[125, 36, 56; 41]]_5 $	20	✔
$ [[125, 70, 26; 15]]_5 $	20	✔
$ [[125, 90, 10; 1]]_5 $	18	✔

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Table 4. Monomials in the support of $ \mathfrak{r}(f_{24}^q) $ for $ q = 5 $

$ j $	$ f_j $	$ \nu(f_j) $	$ \nu(f_j)\bmod{(q^2-1)} $
$ 108 $	$ x^{21}y^2 $	$ 117 $	$ 21 $
$ 84 $	$ x^{15}y^3 $	$ 93 $	$ 21 $
$ 36 $	$ x^9 $	$ 45 $	$ 21 $
$ 12 $	$ x^3y $	$ 21 $	$ 21 $

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Table 5. Results of the modified algorithm for $ q = 3 $

$ f_i $	$ \nu(f_i) $	$ \mathfrak{r}(f_i^q) $	$ \nu(\mathfrak{r}(f_i^q)) $	$ \phi_i $	$ \nu(\phi_i) $
$ 1 $	0		0		0
$ x $	3		9		9
$ y $	4	$ x^4+2y $	12	$ x^4+2y $	12
$ x^2 $	6		18		18
$ xy $	7	$ x^7+2x^3y $	21	$ x^7+2x^3y $	21
$ y^2 $	8		24		24
$ x^3 $	9		3		3
$ x^2y $	10		22		22
$ xy^2 $	11		25		25
$ x^4 $	12	$ x^4 $	12	$ y $	4
$ x^3y $	13		15		15
$ x^2y^2 $	14		26		26
$ x^5 $	15	$ x^7 $	21	$ x^3y $	13

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Table 6. Hermitian EAQECCs exceding the GV bound

$ q $	$ n $	$ c $
2	8	0–3
3	27	1–16
4	64	3–45
5	125	4–96
7	343	10–288
8	512	9–441
9	729	14–640
11	1331	38–1200
13	2197	51–2016
16	4096	45–3825

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